Related papers: Constructing a polynomial whose nodal set is any p…

The number of $n$-gaussoids is shown to be a double exponential function in $n$. The necessary bounds are achieved by studying construction methods for gaussoids that rely on prescribing $3$-minors and encoding the resulting combinatorial…

A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy classes of knots. This operation endows $\mathcal{S}$ with a directed graph structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots in…

We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding…

Factorization machines and polynomial networks are supervised polynomial models based on an efficient low-rank decomposition. We extend these models to the multi-output setting, i.e., for learning vector-valued functions, with application…

This article provides a method for constructing invariants and semi-invariants of a binary $N$-ic form over a field $k$ characteristics $0$ or $p > N$. A practical and broadly applicable sufficient condition for ensuring nontriviality of…

We construct ribbon surfaces of Euler characteristic one for several infinite families of alternating 3-braid closures. We also use a twisted Alexander polynomial obstruction to conclude the classification of smoothly slice knots which are…

Reducing the conditions under which a given set satisfies the stipulations of the subset sum proposition to a set of linear relationships, the question of whether a set satisfies subset sum may be answered in a polynomial number of steps by…

Let L be an algebraic set and let g : R^(n+1) \times L --> R^(2n) (n is even) be a polynomial mapping such that for each l in L there is r(l)>0 such that the mapping g_l = g(.,l) restricted to the sphere S^n(r) is an immersion for every…

An inscribed knot is formed by polygonally connecting points lying on a knot $\gamma$ in parametric order, then closing the path by connecting the first and final points. The stick-knot number of a knot type K is the minimum number of line…

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $\sigma$ be a generator of…

Given a knot in $S^3$, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in $S^{3}$ can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the…

It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various…

Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid…

This paper solves a long standing open problem of whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine by showing that the indistinguishable binomial decision tree can be formed in a 3-SAT…

In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a…

Let n and d be positive integers, let k be a field and let P(n,d;k) be the space of the polynomials in n variables of degree at most d with coefficients in k. Let B(n,d) be the set of the Bernstein-Sato polynomials of all polynomials in…

For a polynomial map $\mathbf{f} : k^n \to k^m$ ($k$ a field), we investigate those polynomials $g \in k[t_1,\ldots, t_n]$ that can be written as a composition $g = h \circ \mathbf{f}$, where $h: k^m \to k$ is an arbitrary function. In the…

In this paper, we propose a new construction of quadratic bent functions in polynomial forms. Right Euclid algorithm in skew-polynomial rings over finite fields of characteristic 2 is applied in the proof.

We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincar\'e-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which…

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the…